Dold--Puppe complexes and exterior powers on K-theory Seminar

It is a well-known fact of homological algebra that the tensor product is not left-exact.

The derived functors of the tensor product, known as Tor modules, fix this problem and provide us with a nice long exact sequence. In the same way we can define left derived functors of any additive functor that is right exact.

Dold--Puppe complexes arise when one wants to define derived functors of non-additive functors, such as the exterior and symmetric powers. Here the usual constructions do not work, due to a failure of homotopy invariance, so one needs to use simplicial methods instead.

In this talk I will review the Dold--Kan correspondence between chain complexes and simplicial R-modules, and use it to define derived functors of exterior powers and other non-additive functors.

I will then describe how I have used the same methods to give a new definition of exterior power operations on higher algebraic K-theory groups.